# Power Transformer Condition-Based Evaluation and Maintenance (CBM) Using Dempster–Shafer Theory (DST)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}represents risk for the i-th component, S

_{i}the inherent cost and H

_{i}the frequency of the fault on the i-th component.

_{i}represents the cost tied to the i-th component, W

_{s}the value of the unit loss in the system downtime state, t

_{i}the time needed for the component to be fixed, L

_{i}the the amount of unit losses and K

_{i}the cost of repairs.

## 2. Methodology

#### 2.1. Maintenance Activities and System Decomposition

- system or system parts, the failure of which can lead to a complete stoppage of the plant;
- system or system parts, the failure of which significantly reduces the overall safety of the plant or may cause the fault of other system parts;
- system or system parts, the failure of which leads to a reduction in the operation level;
- system or system parts, the failure of which affects the reduction of the efficiency of the main production process;
- system or system parts whose failure does not have a direct impact on the production process.

- maintenance carried out by the first free person;
- maintenance carried out within thirty minutes;
- maintenance carried out in the first free period;
- maintenance carried out within 3 months.

#### 2.2. Condition Assessment Equipment Maintenance

#### 2.2.1. Baseline Condition Assessment Maintenance Model

#### 2.2.2. Data Acquisition

## 3. Evidential Reasoning Algorithm

#### 3.1. Maintenance Activities and System Decomposition

- forty percent sure that the gas concentration in the oil is at an average level and 50% sure that the gas concentration is at a very good level;
- completely confident in the assessment that the level of moisture in the oil is very good;
- fifty percent sure that the age level of the oil is at an average level and 50% sure that it is at a very good level.

#### 3.2. System Condition Assessment

_{i}(i = 1, …L) and that they are all connected with general attribute Y. In this case, it is possible to define a set of baseline attributes

_{1}, …e

_{i}… e

_{L}}.

_{1}, …ω

_{i}, …ω

_{L}} where ω

_{i}represents the relative weight of the i-th baseline attribute e

_{i}with a value between 0 and 1 (0 ≤ ω

_{i}≤ 1).

_{1}, …H

_{n}, …H

_{N}},

_{n+1}is higher, i.e., it represents a better condition than H

_{n}, therefore, set H represents an ordered set of elements, starting with the element with the lowest to the one with the highest value. Thus, the evaluation of this element of the set of baseline attributes can be determined with the following:

_{i}) = {(H

_{n}, β

_{n,i}), n = 1,…N} i = 1,…, L;

_{n,i}represents the confidence rating where β

_{n,i}≥ 0, $\sum}_{n=1}^{N}{\beta}_{n,i}\le 1$. If $\sum}_{n=1}^{N}{\beta}_{n,i}=1$, then the condition assessment S(e

_{i}) is considered complete. Otherwise, if $\sum}_{n=1}^{N}{\beta}_{n,i}<1$, then the condition assessment of the given object S(e

_{i}) is considered incomplete.

_{i}. The partial or complete absence of information about a particular attribute required for decision making is not an uncommon phenomenon. In this case, how incomplete information is handled is very important.

_{n}be an evaluation rating and β

_{n}the confidence rating (degree of belief in rating H

_{n}). In this case, it is necessary to calculate the ratings H

_{n}, confidence rating β

_{n}of the general attributes, so that the condition estimates of all the corresponding baseline attributes e

_{i}are considered. The process of calculating the rating and confidence ratings for a general attribute based on information related to baseline attributes is called the aggregation process. For this purpose, the following algorithm is used.

_{n,i}be a baseline attribute weight probability, i.e. value which represents the degree of which the i-th baseline attribute e

_{i}supports the judgment that the baseline attribute y can be estimated by a predefined rating H

_{n}. Furthermore, we can assume that m

_{H,i}represents the remainder of the weight probability, i.e., unassigned probability given all assigned ratings N for the given attribute e

_{i}. The calculation of the weighted probabilities is given by the following:

_{n,i}= ω

_{i}β

_{n,i}n = 1,…, N;

_{i}represents the value resulting from the normalization of the weights of the baseline attributes. The process of normalizing the weights of the base attributes is described in the next section. The remainder of the weighting probability is calculated according to the following:

_{I}

_{(i)}represents the subset of the first i attributes E

_{I}

_{(i)}= {e

_{1},e

_{2},…, e

_{i}} and the weight probability m

_{n}

_{,I(i)}is consequently defined as a rating in which all i attributes support the judgment of the attribute y being estimated at H

_{n}. Furthermore, m

_{H}

_{,I(i)}represents the remainder of the weight probability that is unassigned after all the baseline attributes of E

_{I}

_{(i)}have been estimated. The weight probabilities m

_{n}

_{,I(i)}, m

_{H}

_{,I(i)}can be calculated for E

_{I}

_{(i)}from the basic weight probability m

_{n}

_{,j}and m

_{H}

_{,j}for every n = 1,…, N, and j = 1,…, i. Considering all the above facts, the original recursive algorithm of evidence-based inference can be presented using the following:

_{I}

_{(i+1)}represents the normalizing coefficient where the condition given by the following equation $\sum}_{n=1}^{N}{m}_{n,I\left(i+1\right)}+{m}_{H,I\left(i+1\right)}=1$ is fulfilled. It is important to point out that the baseline attributes E

_{I}

_{(i)}are arbitrarily arranged and that their initial values amount to m

_{n}

_{,I(1)}= m

_{n}

_{,1}and m

_{H}

_{,I(1)}= m

_{H}

_{,1}. Finally, in the original proof algorithm, the combined degree of confidence for the general attribute β

_{n}is given by the following:

_{H}represents the degree of uncertainty or degree of estimation incompleteness.

#### 3.3. Improved Evidential Reasoning Algorithm

#### 3.3.1. Baseline evidential reasoning algorithm upgrade

_{n}if none of the baseline attributes of set E is rated as H

_{n}. This axiom is also referred to as the independence axiom. It refers to the instance, if β

_{n,i}= 0 for every i = 1,…, L, then β

_{n}= 0.

_{n}if all baseline attributes of set E are also accurately graded by H

_{n}. This axiom is also referred to as the consensus axiom. It refers to the instance, if β

_{k}

_{,i}= 1 and β

_{n}

_{,i}= 0 for every I = 1,…, L and n = 1,…, N, n ≠ k, then β

_{k}= 1 i β

_{n}= 0 (n = 1,… N, n ≠ k).

_{H}

_{,i}, shown in (14), is divided into two parts:

_{i}is zero or ${\omega}_{i}$ = 0, ${\overline{m}}_{H,i}$ has a value of 1. Otherwise, if baseline attribute e

_{i}dominates the assessment or ${\omega}_{i}$ = 1, then ${\overline{m}}_{H,i}$ has a value of 0. In simpler terms, ${\overline{m}}_{H,i}$ represents the degree of which other attributes take part in the general assessment.

_{i}). If the assessment of the baseline attribute S(e

_{i}) is complete, then ${\tilde{m}}_{H,i}$ has a value of 0, otherwise, if S(e

_{i}) is incomplete, ${\tilde{m}}_{H,i}$ has a value proportional to ${\omega}_{i}$ and will be between 0 and 1.

_{n}represents the confidence rating for grade H

_{n}to which it is assessed, while β

_{H}represents the unassigned confidence rating and shows the incompleteness in the whole assessment process. It is possible to prove that the combined confidence ratings obtained in the aforementioned way satisfy all four synthesis axioms.

#### 3.3.2. Expected Final Assessment of the Object Condition

_{n}) represents the expected final assessment rating H

_{n}, given the fact that u(H

_{n+1}) > u(H

_{n}) where H

_{n+1}represents a more desirable assessment rating than H

_{n}. Expected final assessment rating u(H

_{n}) can be computed using the probability assignment method or a regression model with partial scores or comparisons. If the final ratings are complete (β

_{H}= 0), the expected final assessment rating of general attribute y can be computed using the following term:

_{n}given by (23) refers to the lower limit at which we can estimate general attribute y. The upper limit of the estimate is given by the plausibility method for H

_{n}or rather (β

_{n}+ β

_{H}). The rating range to which general attribute y can be assessed is given by the interval $\left[{\beta}_{n},{\beta}_{n}+{\beta}_{H}\right]$. If the assessment of the observed object is complete, the interval is set to value β

_{n}; specifically, the interval of confidence ratings depends on the unassigned degree of confidence ratings β

_{H}. In any case, the value to which general attribute y can be assessed is found in the interval β

_{n}to (β

_{n}+ β

_{H}). According to the abovementioned, it is possible to define three values that unambiguously characterize the assessment of general attribute y, the largest, smallest and average values of the expected final assessment rating, which are given by the following:

_{H}= 0, the relation u(y) = u

_{max}(y) = u

_{min}(y) = u

_{avg}(y) is true.

_{i}and a

_{k}based on their final ratings and corresponding intervals, it can be said that the rating of object a

_{i}is more desirable than the rating of object a

_{k}if and only if u

_{min}(y(a

_{i})) > u

_{max}(y(a

_{k})). Two objects have the same assessment if and only if u

_{min}(y(a

_{i})) = u

_{max}(y(a

_{k})). In every other case, comparing the assessments of two objects is incomplete and unreliable. In order to increase the reliability of the comparison of two or more objects, it is necessary to increase the quality of the initial estimates in such a way as to reduce the incompleteness in the attribute estimates for the object conditions a

_{i}and a

_{k}.

#### 3.4. Input Data Analysis

#### 3.4.1. Continuous Variable as Input Data

^{2}(mean value and standard deviation). The question as to how to transform the measured value into a qualitative assessment arises [1].

_{n}, mean value ${\overline{X}}_{n}$ and standard deviation ${\sigma}_{n}^{2}$. Parameters ${\overline{X}}_{n}$ and ${\sigma}_{n}^{2}$ are estimated differently depending on the type of measured quantity, type of device, manufacturer’s recommendation, fault statistics and the experience of the assessor.

_{n}, with a certain degree of belief β

_{n}is added. The degree of belief β

_{n}is defined by the Gaussian distribution with parameters ${\overline{X}}_{n}$ i ${\sigma}_{n}^{2}$:

_{n}is unified to a value of 1:

_{0}= 7 ms. The maximal deviation time for a pole is Δt = 2 ms. The variable that is observed and evaluated is therefore the absolute value of measured time t and ideal time t

_{0}:

#### 3.4.2. Exploitation Time as Input Data

_{n}, n = 1, …, N.

_{n}is transcribed to the time axis:

_{n}is added. The confidence rating is higher as time t approaches the middle of interval t

_{n}. Consequently, each interval n = 1, …, N is associated with a normal distribution with expectancy μ

_{n}= t

_{n}:

_{n}is chosen so that the following conditions are met:

_{H}represents the measurement ambiguity, which can arise as a consequence of disregarding certain factors such as specific component age or the exact amount of malfunction occurrence on such devices.

#### 3.4.3. Discrete Variable as Input Data

_{k}being:

- a specific numerical value (e.g., number of surge arrester operations);
- one of the conditions from a finite set of possible conditions (e.g., condition of a Buchholz relay: A—normal, B—warning and C—trip);
- a descriptive value (e.g., good, bad, average).

_{k}is associated with the set of qualitative grades with the corresponding confidence ratings:

_{nk}for a specific type of system component is determined by the expert assessor or a group of them. Determination methods are mainly based on empirical rules that are defined for each case separately.

#### 3.5. Decomposition Model and Transformer Condition Assessment

_{i}from a set of baseline attributes E should have a defined weight ω

_{i}and a confidence rating βi regarding the grade H

_{n}.

_{i}of the given model have been assigned their appropriate weights ω

_{i}. For the baseline attributes, values for which appropriate measurements can be carried out or be estimated for which confidence rating βi can be determined to its respectable grade H

_{n}. Since this is an extremely complex technical system consisting of several subsystems, determining the decomposition model is not a simple task. By creating a decomposition model, an attempt was made to decompose the power transformer into as many independent baseline elements of the system as possible, the condition of which can be determined by measurement or evaluation. The system elements in the decomposition model are called system attributes (3), and in the following text, the term system attribute will refer to the element of the decomposed system.

#### 3.5.1. Baseline Attributes

#### 3.5.2. Condition Assessment

_{n}, its baseline attributes e

_{i}, respecting their weights ω

_{i}and performing the aggregation procedure as shown in Equations from (18–25). Condition assessment of the observed general attribute is then represented by a probability distribution, i.e., by the confidence rating in the assessments from set H

_{n}as shown in (25). For the condition comparison of two or more points in time, the terms from (26–28) can be used.

_{n}for baseline attributes e

_{i}and their assigned weights ω

_{i}, and after carrying out the aggregation procedure for the second level general attributes, the following distributions are obtained:

S (Load) = {(excellent, 1)} S (Remaining life cycle) = {(excellent, 1)} S (Aging momentum) = {(excellent, 1)} S (General derived attributes) = {(excellent, 1)} S (Winding capacity) = {(excellent, 1)} S (Magnetizing current) = {(excellent, 1)} S (Winding active resistance) = {(excellent, 1)} S (Winding shift control) = {(excellent, 1)} S (Windings) = {(excellent, 1)} S (Chromatography) = {(unsatisfactory, 0.0295), (very good, 0.2489), (excellent, 0.7216)} S (Physical values) = (very good, 0.0454), (excellent, 0.08275), (H, 0.1271)} S (Electric values) = {(excellent, 0.7895), (H, 0.2105)} S (Liquid/solid isolation) = {(unsatisfactory, 0.0084), (very good, 0.0823), (excellent, 0.8306), (H, 0.0787)} S (Boiler air dryer) = {(excellent, 1)} S (Boiler oil level) = {(excellent, 1)} S (Seal loosening) = {(excellent, 1)} S (Boiler) = {(excellent, 1)} S (Conductor insulation dielectric loss factor) = {(good, 0.0563), (very good, 0.2550), (excellent, 0.6886)} S (Conductor insulation capacity) = {(excellent, 1)} S (Conductor thermal imaging) = {(H, 1)} S (Primary side capacity shift index) = {(very good, 0.2951), (excellent, 0.7409)} S (Secondary side capacity shift index) = {(very good, 0.1212), (excellent, 0.8788)} S (Conductors) = {(good, 0.0035), (very good, 0.1492), (excellent, 0.7920), (H, 0.0554)} S (Number of operations) = {(excellent, 1)} S (Total switched current) = {(excellent, 1)} S (Tap changer motor power) = {(excellent, 1)} S (Tap changer) = {(excellent, 1)} S (Pump condition) = {(excellent, 1)} S (Ventilator condition) = {(excellent, 1)} S (Cooler efficiency) = {(excellent, 0.5355), (H, 0.4645)} S (Cooling system) = {(excellent, 0.9394), (H, 0.0606)} which results with the following distribution: S (ET) _{Seventh/May} = {(unsatisfactory, 0.0019), (good, 0.0005), (very good, 0.0385), (excellent, 0.9331), (H, 0.0261)} |

- u (1) = 0;
- u (2) = 0.35;
- u (3) = 0.55;
- u (4) = 0.85
- u (5) = 1.

_{Seventh/May})

_{min}= 0.9661

_{Seventh/May})

_{max}= 0.9921

_{Seventh/May})

_{avg}= 0.9791

_{Seventh/May}) = u((ET)

_{Seventh/May})

_{avg}± Δu = 0.9791 ± 0.013

_{avg}, u

_{min}, u

_{max}) are clearly visible.

#### 3.6. Analysis Results and Uncertainty Review

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Condition\ Grade | H_{1} | H_{2} | … | H_{n} | H_{N} | |
---|---|---|---|---|---|---|

x_{1} | β_{11} | β_{21} | … | β_{n1} | … | β_{N1} |

x_{2} | β_{12} | β_{22} | … | β_{n2} | … | β_{N2} |

… | … | … | … | … | ||

x_{k} | β_{1k} | β_{2k} | β_{nk} | … | β_{Nk} | |

… | … | … | … | … | ||

x_{K} | β_{1K} | β_{2K} | … | β_{nK} | … | β_{NK} |

# ^{1} | Baseline Attribute Type | Description | Unit | Collection Means | Weight |
---|---|---|---|---|---|

1. | Load | Type 2 | ${\omega}_{11}=0.33$ | ||

2. | Remaining life span | year | Type 2 | ${\omega}_{12}=0.33$ | |

3. | Aging momentum | Type 2 | ${\omega}_{13}=0.33$ | ||

4. | C1 ((HV+MV):STN) | Winding capacity in the measured connection (HV + MV):STN | pF | Type 1 | ${\omega}_{211}=0.10$ |

7. | I1 (HV Phase U) | Magnetizing current for HV Phase U | mA | Type 1 | ${\omega}_{221}=0.166$ |

13. | R1 (HV: 1U-N) | Winding active resistance between HV: 1U-N | R/Ω | Type 1 | ${\omega}_{231}=0.143$ |

20. | Asymmetry Lx1 | Dissipative inductance of winding pairs with tap changer position being 1 | % | Type 1 | ${\omega}_{241}=0.33$ |

23. | Hydrogen | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{311}=0.40$ |

24. | Methane | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{314}=0.05$ |

25. | Acetylene | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{313}=0.40$ |

26. | Ethylene | Gases dissolved in transformer oil concentration | μl/L (ppm) | Type 1/2 | ${\omega}_{314}=0.05$ |

27. | Ethane | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{315}=0.05$ |

28. | Carbon monoxide | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{316}=0.05$ |

29. | Carbon dioxide | Gases dissolved in transformer oil concentration | μL/L (ppm) | Type 1/2 | ${\omega}_{317}=0.05$ |

30. | Oil moisture | Gases dissolved in transformer oil concentration | mg/kg | Type 1/2 | ${\omega}_{321}=0.20$ |

31. | Borderline surface tension | Borderline surface tension at 20 °C | mN/m | Type 1 | ${\omega}_{322}=0.20$ |

32. | Paper moisture | % | Type 2 | ${\omega}_{323}=0.20$ | |

35. | Oil temperature in the boiler cover | °C | Type 2 | ${\omega}_{324}=0.20$ | |

36. | Hot-spot oil temperature | °C | Type 2 | ${\omega}_{325}=0.20$ | |

37. | R1 ((HV + MV):STN) | Insulation resistance of the winding and the core between (HV + MV):STN | GΩ (10min) | Type 1 | ${\omega}_{3311}=0.33$ |

40. | tan δ1 ((HV + MV):STN) | Winding insulation Dielectric loss factor between (HV + MV):STN | % | Type 1 | ${\omega}_{3321}=0.33$ |

43. | Oil dielectric strength | Physicochemical tests of transformer oil | kV | Type 1 | ${\omega}_{333}=0.30$ |

44. | Boiler air dryer | Type 1 | ${\omega}_{41}=0.33$ | ||

45. | Boiler oil level | Type 1 | ${\omega}_{42}=0.33$ | ||

46. | Seal loosening | Type 1 | ${\omega}_{43}=0.33$ | ||

47. | tan δ1 (N) | Conductor dielectric loss factor (N) | % | Type 1 | ${\omega}_{511}=0.143$ |

52. | tan δ6 (2V) | Conductor dielectric loss factor (2V) | % | Type 1 | ${\omega}_{516}=0.143$ |

53. | tan δ7 (W) | Conductor dielectric loss factor (2W) | % | Type 1 | ${\omega}_{517}=0.143$ |

54. | C1 (N) | Conductor insulation capacity (Ground point) | pF | Type 1 | ${\omega}_{521}=0.143$ |

55. | C2 (1U) | Conductor insulation capacity (Phase) | pF | Type 1 | ${\omega}_{522}=0.143$ |

61. | Conductor thermal imaging | Type 2 | ${\omega}_{53}=0.10$ | ||

62. | Phase shift index 1U | Primary side conductor capacity change index (1U) | Type 2 | ${\omega}_{541}=0.33$ | |

68. | Number of tap changer operations | Type 2 | ${\omega}_{61}=0.25$ | ||

69. | Total switched current | kA | Type 2 | ${\omega}_{62}=0.25$ | |

70. | Tap changer motor power | % | Type 2 | ${\omega}_{63}=0.40$ | |

71. | Tap changer boiler air dryer | Type 1 | ${\omega}_{64}=0.10$ | ||

72. | Pump 1 condition | Type 2 | ${\omega}_{711}=0.20$ | ||

77. | Ventilator 1 condition | Type 2 | ${\omega}_{721}=0.10$ | ||

87. | Cooler 1 efficiency | % | Type 2 | ${\omega}_{731}=0.20$ | |

91. | Cooler 5 efficiency | % | Type 2 | ${\omega}_{735}=0.20$ |

^{1}Some attributes from the same data spectrum are omitted in order to preserve space.

# | Year | Month | Rating 1 | Rating 2 | Rating 3 | Rating 4 | Rating 5 | Uncertainty | Uavg | Umin | Umax |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | First | August | 0.0000 | 0.0000 | 0.0331 | 0.0592 | 0.3420 | 0.5656 | 0.6934 | 0.4106 | 0.9762 |

2 | Second | February | 0.0000 | 0.0000 | 0.0304 | 0.0244 | 0.4030 | 0.5422 | 0.7116 | 0.4405 | 0.9827 |

3 | Second | August | 0.0000 | 0.0000 | 0.0586 | 0.0000 | 0.3814 | 0.5600 | 0.6936 | 0.4136 | 0.9736 |

4 | Third | February | 0.0000 | 0.0000 | 0.0665 | 0.0000 | 0.3503 | 0.5832 | 0.6785 | 0.3869 | 0.9701 |

5 | Third | August | 0.0000 | 0.0000 | 0.0486 | 0.0184 | 0.7001 | 0.2329 | 0.8589 | 0.7425 | 0.9754 |

6 | Forth | February | 0.0000 | 0.0000 | 0.0059 | 0.0600 | 0.3285 | 0.6056 | 0.6855 | 0.3827 | 0.9884 |

7 | Forth | September | 0.0000 | 0.0057 | 0.0643 | 0.0620 | 0.2727 | 0.5954 | 0.6604 | 0.3627 | 0.9581 |

8 | Fifth | September | 0.0000 | 0.0046 | 0.0000 | 0.0508 | 0.4003 | 0.5443 | 0.7172 | 0.4451 | 0.9894 |

9 | Sixth | February | 0.0000 | 0.0025 | 0.0045 | 0.0350 | 0.7651 | 0.1930 | 0.8946 | 0.7981 | 0.9911 |

10 | Sixth | May | 0.0000 | 0.0030 | 0.0000 | 0.0211 | 0.7525 | 0.2234 | 0.8832 | 0.7714 | 0.9949 |

11 | Sixth | September | 0.0000 | 0.0023 | 0.0006 | 0.0190 | 0.7843 | 0.1938 | 0.8985 | 0.8016 | 0.9954 |

12 | Sixth | December | 0.0026 | 0.0000 | 0.0027 | 0.0103 | 0.7724 | 0.2119 | 0.8886 | 0.7827 | 0.9946 |

13 | Seventh | May | 0.0019 | 0.0000 | 0.0005 | 0.0385 | 0.9331 | 0.0261 | 0.9791 | 0.9661 | 0.9922 |

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## Share and Cite

**MDPI and ACS Style**

Blažević, D.; Keser, T.; Glavaš, H.; Noskov, R.
Power Transformer Condition-Based Evaluation and Maintenance (CBM) Using Dempster–Shafer Theory (DST). *Appl. Sci.* **2023**, *13*, 6731.
https://doi.org/10.3390/app13116731

**AMA Style**

Blažević D, Keser T, Glavaš H, Noskov R.
Power Transformer Condition-Based Evaluation and Maintenance (CBM) Using Dempster–Shafer Theory (DST). *Applied Sciences*. 2023; 13(11):6731.
https://doi.org/10.3390/app13116731

**Chicago/Turabian Style**

Blažević, Damir, Tomislav Keser, Hrvoje Glavaš, and Robert Noskov.
2023. "Power Transformer Condition-Based Evaluation and Maintenance (CBM) Using Dempster–Shafer Theory (DST)" *Applied Sciences* 13, no. 11: 6731.
https://doi.org/10.3390/app13116731